Identifier
Identifier
Mp00096: Binary words Foata bijectionBinary words
Images
0 => 0
1 => 1
00 => 00
01 => 01
10 => 10
11 => 11
000 => 000
001 => 001
010 => 100
011 => 011
100 => 010
101 => 101
110 => 110
111 => 111
0000 => 0000
0001 => 0001
0010 => 1000
0011 => 0011
0100 => 0100
0101 => 1001
0110 => 1010
0111 => 0111
1000 => 0010
1001 => 0101
1010 => 1100
1011 => 1011
1100 => 0110
1101 => 1101
1110 => 1110
1111 => 1111
00000 => 00000
00001 => 00001
00010 => 10000
00011 => 00011
00100 => 01000
00101 => 10001
00110 => 10010
00111 => 00111
01000 => 00100
01001 => 01001
01010 => 11000
01011 => 10011
01100 => 01010
01101 => 10101
01110 => 10110
01111 => 01111
10000 => 00010
10001 => 00101
10010 => 10100
10011 => 01011
10100 => 01100
10101 => 11001
10110 => 11010
10111 => 10111
11000 => 00110
11001 => 01101
11010 => 11100
11011 => 11011
11100 => 01110
11101 => 11101
11110 => 11110
11111 => 11111
000000 => 000000
000001 => 000001
000010 => 100000
000011 => 000011
000100 => 010000
000101 => 100001
000110 => 100010
000111 => 000111
001000 => 001000
001001 => 010001
001010 => 110000
001011 => 100011
001100 => 010010
001101 => 100101
001110 => 100110
001111 => 001111
010000 => 000100
010001 => 001001
010010 => 101000
010011 => 010011
010100 => 011000
010101 => 110001
010110 => 110010
010111 => 100111
011000 => 001010
011001 => 010101
011010 => 110100
011011 => 101011
011100 => 010110
011101 => 101101
011110 => 101110
011111 => 011111
100000 => 000010
100001 => 000101
100010 => 100100
100011 => 001011
100100 => 010100
100101 => 101001
100110 => 101010
100111 => 010111
101000 => 001100
101001 => 011001
101010 => 111000
101011 => 110011
101100 => 011010
101101 => 110101
101110 => 110110
101111 => 101111
110000 => 000110
110001 => 001101
110010 => 101100
110011 => 011011
110100 => 011100
110101 => 111001
110110 => 111010
110111 => 110111
111000 => 001110
111001 => 011101
111010 => 111100
111011 => 111011
111100 => 011110
111101 => 111101
111110 => 111110
111111 => 111111
0000000 => 0000000
0000001 => 0000001
0000010 => 1000000
0000011 => 0000011
0000100 => 0100000
0000101 => 1000001
0000110 => 1000010
0000111 => 0000111
0001000 => 0010000
0001001 => 0100001
0001010 => 1100000
0001011 => 1000011
0001100 => 0100010
0001101 => 1000101
0001110 => 1000110
0001111 => 0001111
0010000 => 0001000
0010001 => 0010001
0010010 => 1010000
0010011 => 0100011
0010100 => 0110000
0010101 => 1100001
0010110 => 1100010
0010111 => 1000111
0011000 => 0010010
0011001 => 0100101
0011010 => 1100100
0011011 => 1001011
0011100 => 0100110
0011101 => 1001101
0011110 => 1001110
0011111 => 0011111
0100000 => 0000100
0100001 => 0001001
0100010 => 1001000
0100011 => 0010011
0100100 => 0101000
0100101 => 1010001
0100110 => 1010010
0100111 => 0100111
0101000 => 0011000
0101001 => 0110001
0101010 => 1110000
0101011 => 1100011
0101100 => 0110010
0101101 => 1100101
0101110 => 1100110
0101111 => 1001111
0110000 => 0001010
0110001 => 0010101
0110010 => 1010100
0110011 => 0101011
0110100 => 0110100
0110101 => 1101001
0110110 => 1101010
0110111 => 1010111
0111000 => 0010110
0111001 => 0101101
0111010 => 1101100
0111011 => 1011011
0111100 => 0101110
0111101 => 1011101
0111110 => 1011110
0111111 => 0111111
1000000 => 0000010
1000001 => 0000101
1000010 => 1000100
1000011 => 0001011
1000100 => 0100100
1000101 => 1001001
1000110 => 1001010
1000111 => 0010111
1001000 => 0010100
1001001 => 0101001
1001010 => 1101000
1001011 => 1010011
1001100 => 0101010
1001101 => 1010101
1001110 => 1010110
1001111 => 0101111
1010000 => 0001100
1010001 => 0011001
1010010 => 1011000
1010011 => 0110011
1010100 => 0111000
1010101 => 1110001
1010110 => 1110010
1010111 => 1100111
1011000 => 0011010
1011001 => 0110101
1011010 => 1110100
1011011 => 1101011
1011100 => 0110110
1011101 => 1101101
1011110 => 1101110
1011111 => 1011111
1100000 => 0000110
1100001 => 0001101
1100010 => 1001100
1100011 => 0011011
1100100 => 0101100
1100101 => 1011001
1100110 => 1011010
1100111 => 0110111
1101000 => 0011100
1101001 => 0111001
1101010 => 1111000
1101011 => 1110011
1101100 => 0111010
1101101 => 1110101
1101110 => 1110110
1101111 => 1101111
1110000 => 0001110
1110001 => 0011101
1110010 => 1011100
1110011 => 0111011
1110100 => 0111100
1110101 => 1111001
1110110 => 1111010
1110111 => 1110111
1111000 => 0011110
1111001 => 0111101
1111010 => 1111100
1111011 => 1111011
1111100 => 0111110
1111101 => 1111101
1111110 => 1111110
1111111 => 1111111
00000000 => 00000000
00000001 => 00000001
00000010 => 10000000
00000011 => 00000011
00000100 => 01000000
00000101 => 10000001
00000110 => 10000010
00000111 => 00000111
00001000 => 00100000
00001001 => 01000001
00001010 => 11000000
00001011 => 10000011
00001100 => 01000010
00001101 => 10000101
00001110 => 10000110
00001111 => 00001111
00010000 => 00010000
00010001 => 00100001
00010010 => 10100000
00010011 => 01000011
00010100 => 01100000
00010101 => 11000001
00010110 => 11000010
00010111 => 10000111
00011000 => 00100010
00011001 => 01000101
00011010 => 11000100
00011011 => 10001011
00011100 => 01000110
00011101 => 10001101
00011110 => 10001110
00011111 => 00011111
00100000 => 00001000
00100001 => 00010001
00100010 => 10010000
00100011 => 00100011
00100100 => 01010000
00100101 => 10100001
00100110 => 10100010
00100111 => 01000111
00101000 => 00110000
00101001 => 01100001
00101010 => 11100000
00101011 => 11000011
00101100 => 01100010
00101101 => 11000101
00101110 => 11000110
00101111 => 10001111
00110000 => 00010010
00110001 => 00100101
00110010 => 10100100
00110011 => 01001011
00110100 => 01100100
00110101 => 11001001
00110110 => 11001010
00110111 => 10010111
00111000 => 00100110
00111001 => 01001101
00111010 => 11001100
00111011 => 10011011
00111100 => 01001110
00111101 => 10011101
00111110 => 10011110
00111111 => 00111111
01000000 => 00000100
01000001 => 00001001
01000010 => 10001000
01000011 => 00010011
01000100 => 01001000
01000101 => 10010001
01000110 => 10010010
01000111 => 00100111
01001000 => 00101000
01001001 => 01010001
01001010 => 11010000
01001011 => 10100011
01001100 => 01010010
01001101 => 10100101
01001110 => 10100110
01001111 => 01001111
01010000 => 00011000
01010001 => 00110001
01010010 => 10110000
01010011 => 01100011
01010100 => 01110000
01010101 => 11100001
01010110 => 11100010
01010111 => 11000111
01011000 => 00110010
01011001 => 01100101
01011010 => 11100100
01011011 => 11001011
01011100 => 01100110
01011101 => 11001101
01011110 => 11001110
01011111 => 10011111
01100000 => 00001010
01100001 => 00010101
01100010 => 10010100
01100011 => 00101011
01100100 => 01010100
01100101 => 10101001
01100110 => 10101010
01100111 => 01010111
01101000 => 00110100
01101001 => 01101001
01101010 => 11101000
01101011 => 11010011
01101100 => 01101010
01101101 => 11010101
01101110 => 11010110
01101111 => 10101111
01110000 => 00010110
01110001 => 00101101
01110010 => 10101100
01110011 => 01011011
01110100 => 01101100
01110101 => 11011001
01110110 => 11011010
01110111 => 10110111
01111000 => 00101110
01111001 => 01011101
01111010 => 11011100
01111011 => 10111011
01111100 => 01011110
01111101 => 10111101
01111110 => 10111110
01111111 => 01111111
10000000 => 00000010
10000001 => 00000101
10000010 => 10000100
10000011 => 00001011
10000100 => 01000100
10000101 => 10001001
10000110 => 10001010
10000111 => 00010111
10001000 => 00100100
10001001 => 01001001
10001010 => 11001000
10001011 => 10010011
10001100 => 01001010
10001101 => 10010101
10001110 => 10010110
10001111 => 00101111
10010000 => 00010100
10010001 => 00101001
10010010 => 10101000
10010011 => 01010011
10010100 => 01101000
10010101 => 11010001
10010110 => 11010010
10010111 => 10100111
10011000 => 00101010
10011001 => 01010101
10011010 => 11010100
10011011 => 10101011
10011100 => 01010110
10011101 => 10101101
10011110 => 10101110
10011111 => 01011111
10100000 => 00001100
10100001 => 00011001
10100010 => 10011000
10100011 => 00110011
10100100 => 01011000
10100101 => 10110001
10100110 => 10110010
10100111 => 01100111
10101000 => 00111000
10101001 => 01110001
10101010 => 11110000
10101011 => 11100011
10101100 => 01110010
10101101 => 11100101
10101110 => 11100110
10101111 => 11001111
10110000 => 00011010
10110001 => 00110101
10110010 => 10110100
10110011 => 01101011
10110100 => 01110100
10110101 => 11101001
10110110 => 11101010
10110111 => 11010111
10111000 => 00110110
10111001 => 01101101
10111010 => 11101100
10111011 => 11011011
10111100 => 01101110
10111101 => 11011101
10111110 => 11011110
10111111 => 10111111
11000000 => 00000110
11000001 => 00001101
11000010 => 10001100
11000011 => 00011011
11000100 => 01001100
11000101 => 10011001
11000110 => 10011010
11000111 => 00110111
11001000 => 00101100
11001001 => 01011001
11001010 => 11011000
11001011 => 10110011
11001100 => 01011010
11001101 => 10110101
11001110 => 10110110
11001111 => 01101111
11010000 => 00011100
11010001 => 00111001
11010010 => 10111000
11010011 => 01110011
11010100 => 01111000
11010101 => 11110001
11010110 => 11110010
11010111 => 11100111
11011000 => 00111010
11011001 => 01110101
11011010 => 11110100
11011011 => 11101011
11011100 => 01110110
11011101 => 11101101
11011110 => 11101110
11011111 => 11011111
11100000 => 00001110
11100001 => 00011101
11100010 => 10011100
11100011 => 00111011
11100100 => 01011100
11100101 => 10111001
11100110 => 10111010
11100111 => 01110111
11101000 => 00111100
11101001 => 01111001
11101010 => 11111000
11101011 => 11110011
11101100 => 01111010
11101101 => 11110101
11101110 => 11110110
11101111 => 11101111
11110000 => 00011110
11110001 => 00111101
11110010 => 10111100
11110011 => 01111011
11110100 => 01111100
11110101 => 11111001
11110110 => 11111010
11110111 => 11110111
11111000 => 00111110
11111001 => 01111101
11111010 => 11111100
11111011 => 11111011
11111100 => 01111110
11111101 => 11111101
11111110 => 11111110
11111111 => 11111111
000000000 => 000000000
000000001 => 000000001
000000010 => 100000000
000000011 => 000000011
000000100 => 010000000
000000101 => 100000001
000000110 => 100000010
000000111 => 000000111
000001000 => 001000000
000001001 => 010000001
000001010 => 110000000
000001011 => 100000011
000001100 => 010000010
000001101 => 100000101
000001110 => 100000110
000001111 => 000001111
000010000 => 000100000
000010001 => 001000001
000010010 => 101000000
000010011 => 010000011
000010100 => 011000000
000010101 => 110000001
000010110 => 110000010
000010111 => 100000111
000011000 => 001000010
000011001 => 010000101
000011010 => 110000100
000011011 => 100001011
000011100 => 010000110
000011101 => 100001101
000011110 => 100001110
000011111 => 000011111
000100000 => 000010000
000100001 => 000100001
000100010 => 100100000
000100011 => 001000011
000100100 => 010100000
000100101 => 101000001
000100110 => 101000010
000100111 => 010000111
000101000 => 001100000
000101001 => 011000001
000101010 => 111000000
000101011 => 110000011
000101100 => 011000010
000101101 => 110000101
000101110 => 110000110
000101111 => 100001111
000110000 => 000100010
000110001 => 001000101
000110010 => 101000100
000110011 => 010001011
000110100 => 011000100
000110101 => 110001001
000110110 => 110001010
000110111 => 100010111
000111000 => 001000110
000111001 => 010001101
000111010 => 110001100
000111011 => 100011011
000111100 => 010001110
000111101 => 100011101
000111110 => 100011110
000111111 => 000111111
001000000 => 000001000
001000001 => 000010001
001000010 => 100010000
001000011 => 000100011
001000100 => 010010000
001000101 => 100100001
001000110 => 100100010
001000111 => 001000111
001001000 => 001010000
001001001 => 010100001
001001010 => 110100000
001001011 => 101000011
001001100 => 010100010
001001101 => 101000101
001001110 => 101000110
001001111 => 010001111
001010000 => 000110000
001010001 => 001100001
001010010 => 101100000
001010011 => 011000011
001010100 => 011100000
001010101 => 111000001
001010110 => 111000010
001010111 => 110000111
001011000 => 001100010
001011001 => 011000101
001011010 => 111000100
001011011 => 110001011
001011100 => 011000110
001011101 => 110001101
001011110 => 110001110
001011111 => 100011111
001100000 => 000010010
001100001 => 000100101
001100010 => 100100100
001100011 => 001001011
001100100 => 010100100
001100101 => 101001001
001100110 => 101001010
001100111 => 010010111
001101000 => 001100100
001101001 => 011001001
001101010 => 111001000
001101011 => 110010011
001101100 => 011001010
001101101 => 110010101
001101110 => 110010110
001101111 => 100101111
001110000 => 000100110
001110001 => 001001101
001110010 => 101001100
001110011 => 010011011
001110100 => 011001100
001110101 => 110011001
001110110 => 110011010
001110111 => 100110111
001111000 => 001001110
001111001 => 010011101
001111010 => 110011100
001111011 => 100111011
001111100 => 010011110
001111101 => 100111101
001111110 => 100111110
001111111 => 001111111
010000000 => 000000100
010000001 => 000001001
010000010 => 100001000
010000011 => 000010011
010000100 => 010001000
010000101 => 100010001
010000110 => 100010010
010000111 => 000100111
010001000 => 001001000
010001001 => 010010001
010001010 => 110010000
010001011 => 100100011
010001100 => 010010010
010001101 => 100100101
010001110 => 100100110
010001111 => 001001111
010010000 => 000101000
010010001 => 001010001
010010010 => 101010000
010010011 => 010100011
010010100 => 011010000
010010101 => 110100001
010010110 => 110100010
010010111 => 101000111
010011000 => 001010010
010011001 => 010100101
010011010 => 110100100
010011011 => 101001011
010011100 => 010100110
010011101 => 101001101
010011110 => 101001110
010011111 => 010011111
010100000 => 000011000
010100001 => 000110001
010100010 => 100110000
010100011 => 001100011
010100100 => 010110000
010100101 => 101100001
010100110 => 101100010
010100111 => 011000111
010101000 => 001110000
010101001 => 011100001
010101010 => 111100000
010101011 => 111000011
010101100 => 011100010
010101101 => 111000101
010101110 => 111000110
010101111 => 110001111
010110000 => 000110010
010110001 => 001100101
010110010 => 101100100
010110011 => 011001011
010110100 => 011100100
010110101 => 111001001
010110110 => 111001010
010110111 => 110010111
010111000 => 001100110
010111001 => 011001101
010111010 => 111001100
010111011 => 110011011
010111100 => 011001110
010111101 => 110011101
010111110 => 110011110
010111111 => 100111111
011000000 => 000001010
011000001 => 000010101
011000010 => 100010100
011000011 => 000101011
011000100 => 010010100
011000101 => 100101001
011000110 => 100101010
011000111 => 001010111
011001000 => 001010100
011001001 => 010101001
011001010 => 110101000
011001011 => 101010011
011001100 => 010101010
011001101 => 101010101
011001110 => 101010110
011001111 => 010101111
011010000 => 000110100
011010001 => 001101001
011010010 => 101101000
011010011 => 011010011
011010100 => 011101000
011010101 => 111010001
011010110 => 111010010
011010111 => 110100111
011011000 => 001101010
011011001 => 011010101
011011010 => 111010100
011011011 => 110101011
011011100 => 011010110
011011101 => 110101101
011011110 => 110101110
011011111 => 101011111
011100000 => 000010110
011100001 => 000101101
011100010 => 100101100
011100011 => 001011011
011100100 => 010101100
011100101 => 101011001
011100110 => 101011010
011100111 => 010110111
011101000 => 001101100
011101001 => 011011001
011101010 => 111011000
011101011 => 110110011
011101100 => 011011010
011101101 => 110110101
011101110 => 110110110
011101111 => 101101111
011110000 => 000101110
011110001 => 001011101
011110010 => 101011100
011110011 => 010111011
011110100 => 011011100
011110101 => 110111001
011110110 => 110111010
011110111 => 101110111
011111000 => 001011110
011111001 => 010111101
011111010 => 110111100
011111011 => 101111011
011111100 => 010111110
011111101 => 101111101
011111110 => 101111110
011111111 => 011111111
100000000 => 000000010
100000001 => 000000101
100000010 => 100000100
100000011 => 000001011
100000100 => 010000100
100000101 => 100001001
100000110 => 100001010
100000111 => 000010111
100001000 => 001000100
100001001 => 010001001
100001010 => 110001000
100001011 => 100010011
100001100 => 010001010
100001101 => 100010101
100001110 => 100010110
100001111 => 000101111
100010000 => 000100100
100010001 => 001001001
100010010 => 101001000
100010011 => 010010011
100010100 => 011001000
100010101 => 110010001
100010110 => 110010010
100010111 => 100100111
100011000 => 001001010
100011001 => 010010101
100011010 => 110010100
100011011 => 100101011
100011100 => 010010110
100011101 => 100101101
100011110 => 100101110
100011111 => 001011111
100100000 => 000010100
100100001 => 000101001
100100010 => 100101000
100100011 => 001010011
100100100 => 010101000
100100101 => 101010001
100100110 => 101010010
100100111 => 010100111
100101000 => 001101000
100101001 => 011010001
100101010 => 111010000
100101011 => 110100011
100101100 => 011010010
100101101 => 110100101
100101110 => 110100110
100101111 => 101001111
100110000 => 000101010
100110001 => 001010101
100110010 => 101010100
100110011 => 010101011
100110100 => 011010100
100110101 => 110101001
100110110 => 110101010
100110111 => 101010111
100111000 => 001010110
100111001 => 010101101
100111010 => 110101100
100111011 => 101011011
100111100 => 010101110
100111101 => 101011101
100111110 => 101011110
100111111 => 010111111
101000000 => 000001100
101000001 => 000011001
101000010 => 100011000
101000011 => 000110011
101000100 => 010011000
101000101 => 100110001
101000110 => 100110010
101000111 => 001100111
101001000 => 001011000
101001001 => 010110001
101001010 => 110110000
101001011 => 101100011
101001100 => 010110010
101001101 => 101100101
101001110 => 101100110
101001111 => 011001111
101010000 => 000111000
101010001 => 001110001
101010010 => 101110000
101010011 => 011100011
101010100 => 011110000
101010101 => 111100001
101010110 => 111100010
101010111 => 111000111
101011000 => 001110010
101011001 => 011100101
101011010 => 111100100
101011011 => 111001011
101011100 => 011100110
101011101 => 111001101
101011110 => 111001110
101011111 => 110011111
101100000 => 000011010
101100001 => 000110101
101100010 => 100110100
101100011 => 001101011
101100100 => 010110100
101100101 => 101101001
101100110 => 101101010
101100111 => 011010111
101101000 => 001110100
101101001 => 011101001
101101010 => 111101000
101101011 => 111010011
101101100 => 011101010
101101101 => 111010101
101101110 => 111010110
101101111 => 110101111
101110000 => 000110110
101110001 => 001101101
101110010 => 101101100
101110011 => 011011011
101110100 => 011101100
101110101 => 111011001
101110110 => 111011010
101110111 => 110110111
101111000 => 001101110
101111001 => 011011101
101111010 => 111011100
101111011 => 110111011
101111100 => 011011110
101111101 => 110111101
101111110 => 110111110
101111111 => 101111111
110000000 => 000000110
110000001 => 000001101
110000010 => 100001100
110000011 => 000011011
110000100 => 010001100
110000101 => 100011001
110000110 => 100011010
110000111 => 000110111
110001000 => 001001100
110001001 => 010011001
110001010 => 110011000
110001011 => 100110011
110001100 => 010011010
110001101 => 100110101
110001110 => 100110110
110001111 => 001101111
110010000 => 000101100
110010001 => 001011001
110010010 => 101011000
110010011 => 010110011
110010100 => 011011000
110010101 => 110110001
110010110 => 110110010
110010111 => 101100111
110011000 => 001011010
110011001 => 010110101
110011010 => 110110100
110011011 => 101101011
110011100 => 010110110
110011101 => 101101101
110011110 => 101101110
110011111 => 011011111
110100000 => 000011100
110100001 => 000111001
110100010 => 100111000
110100011 => 001110011
110100100 => 010111000
110100101 => 101110001
110100110 => 101110010
110100111 => 011100111
110101000 => 001111000
110101001 => 011110001
110101010 => 111110000
110101011 => 111100011
110101100 => 011110010
110101101 => 111100101
110101110 => 111100110
110101111 => 111001111
110110000 => 000111010
110110001 => 001110101
110110010 => 101110100
110110011 => 011101011
110110100 => 011110100
110110101 => 111101001
110110110 => 111101010
110110111 => 111010111
110111000 => 001110110
110111001 => 011101101
110111010 => 111101100
110111011 => 111011011
110111100 => 011101110
110111101 => 111011101
110111110 => 111011110
110111111 => 110111111
111000000 => 000001110
111000001 => 000011101
111000010 => 100011100
111000011 => 000111011
111000100 => 010011100
111000101 => 100111001
111000110 => 100111010
111000111 => 001110111
111001000 => 001011100
111001001 => 010111001
111001010 => 110111000
111001011 => 101110011
111001100 => 010111010
111001101 => 101110101
111001110 => 101110110
111001111 => 011101111
111010000 => 000111100
111010001 => 001111001
111010010 => 101111000
111010011 => 011110011
111010100 => 011111000
111010101 => 111110001
111010110 => 111110010
111010111 => 111100111
111011000 => 001111010
111011001 => 011110101
111011010 => 111110100
111011011 => 111101011
111011100 => 011110110
111011101 => 111101101
111011110 => 111101110
111011111 => 111011111
111100000 => 000011110
111100001 => 000111101
111100010 => 100111100
111100011 => 001111011
111100100 => 010111100
111100101 => 101111001
111100110 => 101111010
111100111 => 011110111
111101000 => 001111100
111101001 => 011111001
111101010 => 111111000
111101011 => 111110011
111101100 => 011111010
111101101 => 111110101
111101110 => 111110110
111101111 => 111101111
111110000 => 000111110
111110001 => 001111101
111110010 => 101111100
111110011 => 011111011
111110100 => 011111100
111110101 => 111111001
111110110 => 111111010
111110111 => 111110111
111111000 => 001111110
111111001 => 011111101
111111010 => 111111100
111111011 => 111111011
111111100 => 011111110
111111101 => 111111101
111111110 => 111111110
111111111 => 111111111
1010101010 => 1111100000
1010101100 => 0111100010
1010110010 => 1011100100
1010110100 => 0111100100
1010111000 => 0011100110
1011001010 => 1101101000
1011001100 => 0101101010
1011010010 => 1011101000
1011010100 => 0111101000
1011011000 => 0011101010
1011100010 => 1001101100
1011100100 => 0101101100
1011101000 => 0011101100
1011110000 => 0001101110
1100101010 => 1110110000
1100101100 => 0110110010
1100110010 => 1010110100
1100110100 => 0110110100
1100111000 => 0010110110
1101001010 => 1101110000
1101001100 => 0101110010
1101010010 => 1011110000
1101010100 => 0111110000
1101011000 => 0011110010
1101100010 => 1001110100
1101100100 => 0101110100
1101101000 => 0011110100
1101110000 => 0001110110
1110001010 => 1100111000
1110001100 => 0100111010
1110010010 => 1010111000
1110010100 => 0110111000
1110011000 => 0010111010
1110100010 => 1001111000
1110100100 => 0101111000
1110101000 => 0011111000
1110110000 => 0001111010
1111000010 => 1000111100
1111000100 => 0100111100
1111001000 => 0010111100
1111010000 => 0001111100
1111100000 => 0000111110
101010101010 => 111111000000
101010101100 => 011111000010
101010110010 => 101111000100
101010110100 => 011111000100
101010111000 => 001111000110
101011001010 => 110111001000
101011001100 => 010111001010
101011010010 => 101111001000
101011010100 => 011111001000
101011011000 => 001111001010
101011100010 => 100111001100
101011100100 => 010111001100
101011101000 => 001111001100
101011110000 => 000111001110
101100101010 => 111011010000
101100101100 => 011011010010
101100110010 => 101011010100
101100110100 => 011011010100
101100111000 => 001011010110
101101001010 => 110111010000
101101001100 => 010111010010
101101010010 => 101111010000
101101010100 => 011111010000
101101011000 => 001111010010
101101100010 => 100111010100
101101100100 => 010111010100
101101101000 => 001111010100
101101110000 => 000111010110
101110001010 => 110011011000
101110001100 => 010011011010
101110010010 => 101011011000
101110010100 => 011011011000
101110011000 => 001011011010
101110100010 => 100111011000
101110100100 => 010111011000
101110101000 => 001111011000
101110110000 => 000111011010
101111000010 => 100011011100
101111000100 => 010011011100
101111001000 => 001011011100
101111010000 => 000111011100
101111100000 => 000011011110
110010101010 => 111101100000
110010101100 => 011101100010
110010110010 => 101101100100
110010110100 => 011101100100
110010111000 => 001101100110
110011001010 => 110101101000
110011001100 => 010101101010
110011010010 => 101101101000
110011010100 => 011101101000
110011011000 => 001101101010
110011100010 => 100101101100
110011100100 => 010101101100
110011101000 => 001101101100
110011110000 => 000101101110
110100101010 => 111011100000
110100101100 => 011011100010
110100110010 => 101011100100
110100110100 => 011011100100
110100111000 => 001011100110
110101001010 => 110111100000
110101001100 => 010111100010
110101010010 => 101111100000
110101010100 => 011111100000
110101011000 => 001111100010
110101100010 => 100111100100
110101100100 => 010111100100
110101101000 => 001111100100
110101110000 => 000111100110
110110001010 => 110011101000
110110001100 => 010011101010
110110010010 => 101011101000
110110010100 => 011011101000
110110011000 => 001011101010
110110100010 => 100111101000
110110100100 => 010111101000
110110101000 => 001111101000
110110110000 => 000111101010
110111000010 => 100011101100
110111000100 => 010011101100
110111001000 => 001011101100
110111010000 => 000111101100
110111100000 => 000011101110
111000101010 => 111001110000
111000101100 => 011001110010
111000110010 => 101001110100
111000110100 => 011001110100
111000111000 => 001001110110
111001001010 => 110101110000
111001001100 => 010101110010
111001010010 => 101101110000
111001010100 => 011101110000
111001011000 => 001101110010
111001100010 => 100101110100
111001100100 => 010101110100
111001101000 => 001101110100
111001110000 => 000101110110
111010001010 => 110011110000
111010001100 => 010011110010
111010010010 => 101011110000
111010010100 => 011011110000
111010011000 => 001011110010
111010100010 => 100111110000
111010100100 => 010111110000
111010101000 => 001111110000
111010110000 => 000111110010
111011000010 => 100011110100
111011000100 => 010011110100
111011001000 => 001011110100
111011010000 => 000111110100
111011100000 => 000011110110
111100001010 => 110001111000
111100001100 => 010001111010
111100010010 => 101001111000
111100010100 => 011001111000
111100011000 => 001001111010
111100100010 => 100101111000
111100100100 => 010101111000
111100101000 => 001101111000
111100110000 => 000101111010
111101000010 => 100011111000
111101000100 => 010011111000
111101001000 => 001011111000
111101010000 => 000111111000
111101100000 => 000011111010
111110000010 => 100001111100
111110000100 => 010001111100
111110001000 => 001001111100
111110010000 => 000101111100
111110100000 => 000011111100
111111000000 => 000001111110
10101010101100 => 01111110000010
10101010110010 => 10111110000100
10101010110100 => 01111110000100
10101011001100 => 01011110001010
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Description
The Foata bijection $\phi$ is a bijection on the set of words of given content (by a slight generalization of Section 2 in [1]).
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$. At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
For instance, to compute $\phi(4154223)$, the sequence of words is
  • 4,
  • |4|1 --> 41,
  • |4|1|5 --> 415,
  • |415|4 --> 5414,
  • |5|4|14|2 --> 54412,
  • |5441|2|2 --> 154422,
  • |1|5442|2|3 --> 1254423.
So $\phi(4154223) = 1254423$.
Properties
bijectiveA map $\phi: A \rightarrow B$ is bijective if it is both injective and surjective. This is if (1) $\phi(a) = \phi(b)$ implies $a=b$, and (2) for any $b \in B$, there is an $a \in A$ such that $\phi(a) = b$., gradedA map $\phi: A \rightarrow B$ is graded for graded sets $A$ and $B$ if $\operatorname{deg}(a) = \operatorname{deg}(a')$ implies $\operatorname{deg}(\phi(a)) = \operatorname{deg}(\phi(a'))$ for all $a, a' \in A$.
References
[1] Foata, D., Sch├╝tzenberger, M.-P. Major index and inversion number of permutations mathscinet:0506852
[2] Mp00067Foata bijection
Code
def foata_bijection(w):
    return w.foata_bijection()